20 problems

A knot $K$ is topologically slice if it bounds a topological slice disk $\Delta$ in $B^4$. It is topologically homotopy ribbon if it bounds such a disk for which the inclusion-indu…

Let $K$ be an even 3-strand pretzel knot, meaning a pretzel knot with an even parameter in the relevant parity convention. Topological sliceness and ribbonness conjecture. Then $K$…

Equivariant slice-ribbon conjecture. An equivariant slice knot is equivariant ribbon.

Knot Floer homology conjecture. The group $\widehat{HFK}(J)$ is supported nontrivially in exactly $g$ diagonals, shifted up in Maslov grading from $K$. Hence the corresponding exot…

Let $C=11n_{34}$ denote the Conway knot. It has Alexander polynomial $1$ and is topologically slice but not smoothly slice. Conway knot ribbon minimality conjecture. The Conway kno…

Let $K\subset S^3=\partial D^4$ be a knot. It is slice if it bounds a smoothly, properly embedded disk in $D^4$; it is ribbon if it bounds a disk whose only singularities are ribbo…

Let $K\subseteq S^3$ be a knot, and let $Wh^\pm(K)$ denote its positive or negative clasped, untwisted Whitehead double. Hedden's Whitehead-doubling conjecture. The knot…

Let $K$ be a knot in $S^3$. Its positive Whitehead double is the satellite knot obtained using the positive Whitehead pattern. Whitehead double sliceness conjecture. The knot $K$ i…

Let $K$ be a knot, let $p$ and $q$ be relatively prime integers with $|p|>1$, and let $K_{p,q}$ denote the $(p,q)$-cable of $K$. Let $\mathcal{C}$ denote the smooth knot concordanc…

Let $K$ be a knot in $S^3$. A concordance is a cobordism of minimal intrinsic complexity, and it has minimal extrinsic complexity when it has no index $2$ Morse critical points. Sl…

Let $K$ be a knot, and let a slice derivative mean a derivative of $K$ that is slice in the relevant category. Generalised Kauffman conjecture. If $K$ is a topologically (respectiv…

Kauffman's conjecture. There exists an essential simple closed curve $d$ on $F$ such that

Let $K$ be a knot with a slice disk $\Delta$, and let $J$ be a derivative of $K$ associated to $\Delta$. Kauffman's conjecture. If $K$ is a slice knot via $\Delta$, then $J$ is a slic…

Let $K$ be the $(p,q)$-cable of a knot $J$. A knot is slice if it bounds a smooth, properly embedded disk in $B^4$. Slice cable conjecture. The knot $K$ is slice if and only if $J$…

Homotopy ribbon-slice conjecture. A knot is slice if and only if it is homotopy ribbon.

Let $K$ be a slice knot. A Seifert surface $F$ stabilizes to a Seifert surface $F'$ if $F'$ is obtained from $F$ by finitely many trivial one-handle attachments. A Levine surface i…

Let $K$ be a slice knot and let $F$ be a genus-one Seifert surface for $K$. A Levine–Tristram signature function is the signature function associated to a knot, denoted in the sour…

Let $K$ be a slice knot, and let $F$ be a genus-one Seifert surface for $K$. For a simple closed curve $d$ on $F$, write $d^+$ for its positive push-off and define its self-linking…

Let $K$ be a slice knot, and let $F$ be a genus-one Seifert surface for $K$. For a simple closed curve $d$ on $F$, write $d^+$ for its positive push-off and…

A knot is a smooth embedding $\mathbb{S}^1\hookrightarrow\mathbb{S}^3=\partial\mathbb{B}^4$. It is slice if it bounds a smoothly embedded disk in $\mathbb{B}^4$. A genus one Seifer…